Speaker
Description
The amplitudes of the rare kaon decays $K\to\pi\ell^+\ell^-$, $(K,\pi) = (K^+,\pi^+), (K_S , \pi^0)$, $\ell = e , \mu$, are dominated by their long-distance component. The latter is given by the exchange of a virtual photon between the lepton pair and the four-quark operators of the $\Delta S =1$ weak effective Lagrangian. The corresponding $K - \pi - \gamma^*$ form factor is described as a sum of three contributions. At large Euclidian virtualities, it behaves as powers of the logarithm of the photon momentum squared. The one-loop correction of this type is easily computed, and the two-loop one is almost entirely determined by a renormalization-group argument combined with existing calculations of the two-loop anomalous dimensions of the $\Delta S =1$ four-quark operators. At long distances, the form factor is written as an unsubtracted dispersion relation. The absorptive part, when restricted to two-pion intermediate states, is given by the product of the electromagnetic form factor of the pion times the $P$-wave projection of the amplitude for $K\pi\to\pi^+\pi^-$ scattering. Finally, the intermediate region is described by an infinite sum of zero-width resonances, with residues tuned such as to reproduce the correct short-distance behaviour. Predictions of the amplitudes based on this description are presented and compared to experiment. Possibilities to improve this phenomenological description are also discussed.
